Carathéodory's existence theorem

Carathéodory's existence theorem Differential equations Navier–Stokes differential equations used to simulate airflow around an obstruction Scope show Fields Classification show Types show Relation to processes Solution show Existence and uniqueness show General topics show Solution methods People show List vte In mathematics, Carathéodory's existence theorem says that an ordinary differential equation has a solution under relatively mild conditions. It is a generalization of Peano's existence theorem. Peano's theorem requires that the right-hand side of the differential equation be continuous, while Carathéodory's theorem shows existence of solutions (in a more general sense) for some discontinuous equations. The theorem is named after Constantin Carathéodory.

Contents 1 Introduction 2 Statement of the theorem 3 Uniqueness of a solution 4 Example 5 See also 6 Notes 7 References Introduction Consider the differential equation {displaystyle y'(t)=f(t,y(t))} with initial condition {displaystyle y(t_{0})=y_{0},} where the function ƒ is defined on a rectangular domain of the form {displaystyle R={(t,y)in mathbf {R} times mathbf {R} ^{n},:,|t-t_{0}|leq a,|y-y_{0}|leq b}.} Peano's existence theorem states that if ƒ is continuous, then the differential equation has at least one solution in a neighbourhood of the initial condition.[1] However, it is also possible to consider differential equations with a discontinuous right-hand side, like the equation {displaystyle y'(t)=H(t),quad y(0)=0,} where H denotes the Heaviside function defined by {displaystyle H(t)={begin{cases}0,&{text{if }}tleq 0;\1,&{text{if }}t>0.end{cases}}} It makes sense to consider the ramp function {displaystyle y(t)=int _{0}^{t}H(s),mathrm {d} s={begin{cases}0,&{text{if }}tleq 0;\t,&{text{if }}t>0end{cases}}} as a solution of the differential equation. Strictly speaking though, it does not satisfy the differential equation at {displaystyle t=0} , because the function is not differentiable there. This suggests that the idea of a solution be extended to allow for solutions that are not everywhere differentiable, thus motivating the following definition.

A function y is called a solution in the extended sense of the differential equation {displaystyle y'=f(t,y)} with initial condition {displaystyle y(t_{0})=y_{0}} if y is absolutely continuous, y satisfies the differential equation almost everywhere and y satisfies the initial condition.[2] The absolute continuity of y implies that its derivative exists almost everywhere.[3] Statement of the theorem Consider the differential equation {displaystyle y'(t)=f(t,y(t)),quad y(t_{0})=y_{0},} with {displaystyle f} defined on the rectangular domain {displaystyle R={(t,y),|,|t-t_{0}|leq a,|y-y_{0}|leq b}} . If the function {displaystyle f} satisfies the following three conditions: {displaystyle f(t,y)} is continuous in {displaystyle y} for each fixed {displaystyle t} , {displaystyle f(t,y)} is measurable in {displaystyle t} for each fixed {displaystyle y} , there is a Lebesgue-integrable function {displaystyle m:[t_{0}-a,t_{0}+a]to [0,infty )} such that {displaystyle |f(t,y)|leq m(t)} for all {displaystyle (t,y)in R} , then the differential equation has a solution in the extended sense in a neighborhood of the initial condition.[4] A mapping {displaystyle fcolon Rto mathbf {R} ^{n}} is said to satisfy the Carathéodory conditions on {displaystyle R} if it fulfills the condition of the theorem.[5] Uniqueness of a solution Assume that the mapping {displaystyle f} satisfies the Carathéodory conditions on {displaystyle R} and there is a Lebesgue-integrable function {displaystyle k:[t_{0}-a,t_{0}+a]to [0,infty )} , such that {displaystyle |f(t,y_{1})-f(t,y_{2})|leq k(t)|y_{1}-y_{2}|,} for all {displaystyle (t,y_{1})in R,(t,y_{2})in R.} Then, there exists a unique solution {displaystyle y(t)=y(t,t_{0},y_{0})} to the initial value problem {displaystyle y'(t)=f(t,y(t)),quad y(t_{0})=y_{0}.} Moreover, if the mapping {displaystyle f} is defined on the whole space {displaystyle mathbf {R} times mathbf {R} ^{n}} and if for any initial condition {displaystyle (t_{0},y_{0})in mathbf {R} times mathbf {R} ^{n}} , there exists a compact rectangular domain {displaystyle R_{(t_{0},y_{0})}subset mathbf {R} times mathbf {R} ^{n}} such that the mapping {displaystyle f} satisfies all conditions from above on {displaystyle R_{(t_{0},y_{0})}} . Then, the domain {displaystyle Esubset mathbf {R} ^{2+n}} of definition of the function {displaystyle y(t,t_{0},y_{0})} is open and {displaystyle y(t,t_{0},y_{0})} is continuous on {displaystyle E} .[6] Example Consider a linear initial value problem of the form {displaystyle y'(t)=A(t)y(t)+b(t),quad y(t_{0})=y_{0}.} Here, the components of the matrix-valued mapping {displaystyle Acolon mathbf {R} to mathbf {R} ^{ntimes n}} and of the inhomogeneity {displaystyle bcolon mathbf {R} to mathbf {R} ^{n}} are assumed to be integrable on every finite interval. Then, the right hand side of the differential equation satisfies the Carathéodory conditions and there exists a unique solution to the initial value problem.[7] See also Mathematics portal Picard–Lindelöf theorem Cauchy–Kowalevski theorem Notes ^ Coddington & Levinson (1955), Theorem 1.2 of Chapter 1 ^ Coddington & Levinson (1955), page 42 ^ Rudin (1987), Theorem 7.18 ^ Coddington & Levinson (1955), Theorem 1.1 of Chapter 2 ^ Hale (1980), p.28 ^ Hale (1980), Theorem 5.3 of Chapter 1 ^ Hale (1980), p.30 References Coddington, Earl A.; Levinson, Norman (1955), Theory of Ordinary Differential Equations, New York: McGraw-Hill. Hale, Jack K. (1980), Ordinary Differential Equations (2nd ed.), Malabar: Robert E. Krieger Publishing Company, ISBN 0-89874-011-8. Rudin, Walter (1987), Real and complex analysis (3rd ed.), New York: McGraw-Hill, ISBN 978-0-07-054234-1, MR 0924157. Categories: Ordinary differential equationsTheorems in analysis